Size quantization of an exciton: A toy model of the "dead layer"

TL;DR

This paper presents a theoretical analysis of exciton size quantization in large nanocrystals, revealing that the boundary reflection phase and a 'dead layer' significantly influence energy levels, especially when the hole is much heavier.

Contribution

It introduces an analytical model for exciton boundary reflection, highlighting the formation of a 'dead layer' where the hole does not reach the boundary, a novel insight into exciton boundary interactions.

Findings

Level positions are independent of Bohr radius for large nanocrystals.

The correction to levels depends on the boundary reflection phase.

A 'dead layer' larger than the Bohr radius forms due to slow hole motion near the boundary.

Abstract

Size-quantization levels of an exciton in large nanocrystals is studied theoretically. For the nanocrystal size, $L$, much bigger than the Bohr radius, $a_B$, the level positions do not depend on $a_B$. The correction to the levels in a small parameter $a_B/L$ depends on the reflection phase of the exciton from the boundary. Calculation of this phase constitutes a three-body problem: electron, hole, and the boundary. This calculation can be performed analytically in the limit when the hole is much heavier than the electron. Physically, a slow motion of the hole towards the boundary takes place in the effective potential created by the fast motion of the electron orbiting the hole and touching the boundary. As a result, the hole is reflected before reaching the boundary. The distance of the closest approach of the hole to the boundary (the dead layer) exceeds $a_B$ parametrically.